Eisenstein series on affine KacMoody groups over function fields
Authors:
KyuHwan Lee and Philip Lombardo
Journal:
Trans. Amer. Math. Soc. 366 (2014), 21212165
MSC (2010):
Primary 22E67; Secondary 11F70
Published electronically:
September 26, 2013
Fulltext PDF
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Abstract: In his pioneering work, H. Garland constructed Eisenstein series on affine KacMoody groups over the field of real numbers. He established the almost everywhere convergence of these series, obtained a formula for their constant terms, and proved a functional equation for the constant terms. In his subsequent paper, the convergence of the Eisenstein series was obtained. In this paper, we define Eisenstein series on affine KacMoody groups over global function fields using an adelic approach. In the course of proving the convergence of these Eisenstein series, we also calculate a formula for the constant terms and prove their convergence and functional equations.
 [1]
Abhay
Ashtekar and Jerzy
Lewandowski, Projective techniques and functional integration for
gauge theories, J. Math. Phys. 36 (1995), no. 5,
2170–2191. MR 1329251
(96a:58033), http://dx.doi.org/10.1063/1.531037
 [2]
Howard
Garland, The arithmetic theory of loop groups, Inst. Hautes
Études Sci. Publ. Math. 52 (1980), 5–136. MR 601519
(83a:20057)
 [3]
Howard
Garland, The arithmetic theory of loop groups. II. The
Hilbertmodular case, J. Algebra 209 (1998),
no. 2, 446–532. MR 1659899
(2000h:20088), http://dx.doi.org/10.1006/jabr.1998.7529
 [4]
H.
Garland, Certain Eisenstein series on loop groups: convergence and
the constant term, Algebraic groups and arithmetic, Tata Inst. Fund.
Res., Mumbai, 2004, pp. 275–319. MR 2094114
(2005i:11118)
 [5]
Howard
Garland, Absolute convergence of Eisenstein series on loop
groups, Duke Math. J. 135 (2006), no. 2,
203–260. MR 2267283
(2008g:22032), http://dx.doi.org/10.1215/S0012709406135214
 [6]
Howard
Garland and James
Lepowsky, Lie algebra homology and the MacdonaldKac formulas,
Invent. Math. 34 (1976), no. 1, 37–76. MR 0414645
(54 #2744)
 [7]
Stephen
S. Gelbart and Stephen
D. Miller, Riemann’s zeta function and
beyond, Bull. Amer. Math. Soc. (N.S.)
41 (2004), no. 1,
59–112. MR
2015450 (2005a:11135), http://dx.doi.org/10.1090/S0273097903009959
 [8]
G.
Harder, Chevalley groups over function fields and automorphic
forms, Ann. of Math. (2) 100 (1974), 249–306.
MR
0563090 (58 #27799)
 [9]
James
E. Humphreys, Linear algebraic groups, SpringerVerlag, New
YorkHeidelberg, 1975. Graduate Texts in Mathematics, No. 21. MR 0396773
(53 #633)
 [10]
N.
Iwahori and H.
Matsumoto, On some Bruhat decomposition and the structure of the
Hecke rings of 𝔭adic Chevalley groups, Inst. Hautes
Études Sci. Publ. Math. 25 (1965), 5–48. MR 0185016
(32 #2486)
 [11]
Victor
G. Kac, Infinitedimensional Lie algebras, 3rd ed., Cambridge
University Press, Cambridge, 1990. MR 1104219
(92k:17038)
 [12]
Shrawan
Kumar, KacMoody groups, their flag varieties and representation
theory, Progress in Mathematics, vol. 204, Birkhäuser
Boston, Inc., Boston, MA, 2002. MR 1923198
(2003k:22022)
 [13]
Robert
P. Langlands, Euler products, Yale University Press, New
Haven, Conn.London, 1971. A James K. Whittemore Lecture in Mathematics
given at Yale University, 1967; Yale Mathematical Monographs, 1. MR 0419366
(54 #7387)
 [14]
Robert
P. Langlands, On the functional equations satisfied by Eisenstein
series, Lecture Notes in Mathematics, Vol. 544, SpringerVerlag,
BerlinNew York, 1976. MR 0579181
(58 #28319)
 [15]
Manish
M. Patnaik, Geometry of loop Eisenstein series, ProQuest LLC,
Ann Arbor, MI, 2008. Thesis (Ph.D.)–Yale University. MR
2711828
 [16]
Vladimir
Platonov and Andrei
Rapinchuk, Algebraic groups and number theory, Pure and
Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994.
Translated from the 1991 Russian original by Rachel Rowen. MR 1278263
(95b:11039)
 [17]
Michael
Rosen, Number theory in function fields, Graduate Texts in
Mathematics, vol. 210, SpringerVerlag, New York, 2002. MR 1876657
(2003d:11171)
 [18]
André
Weil, Adeles and algebraic groups, Progress in Mathematics,
vol. 23, Birkhäuser, Boston, Mass., 1982. With appendices by M.
Demazure and Takashi Ono. MR 670072
(83m:10032)
 [1]
 Abhay Ashtekar and Jerzy Lewandowski, Projective techniques and functional integration for gauge theories, J. Math. Phys. 36 (1995), no. 5, 21702191. MR 1329251 (96a:58033), http://dx.doi.org/10.1063/1.531037
 [2]
 Howard Garland, The arithmetic theory of loop groups, Inst. Hautes Études Sci. Publ. Math. 52 (1980), 5136. MR 601519 (83a:20057)
 [3]
 Howard Garland, The arithmetic theory of loop groups. II. The Hilbertmodular case, J. Algebra 209 (1998), no. 2, 446532. MR 1659899 (2000h:20088), http://dx.doi.org/10.1006/jabr.1998.7529
 [4]
 H. Garland, Certain Eisenstein series on loop groups: convergence and the constant term, Algebraic groups and arithmetic, Tata Inst. Fund. Res., Mumbai, 2004, pp. 275319. MR 2094114 (2005i:11118)
 [5]
 Howard Garland, Absolute convergence of Eisenstein series on loop groups, Duke Math. J. 135 (2006), no. 2, 203260. MR 2267283 (2008g:22032), http://dx.doi.org/10.1215/S0012709406135214
 [6]
 Howard Garland and James Lepowsky, Lie algebra homology and the MacdonaldKac formulas, Invent. Math. 34 (1976), no. 1, 3776. MR 0414645 (54 #2744)
 [7]
 Stephen S. Gelbart and Stephen D. Miller, Riemann's zeta function and beyond, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 1, 59112. MR 2015450 (2005a:11135), http://dx.doi.org/10.1090/S0273097903009959
 [8]
 G. Harder, Chevalley groups over function fields and automorphic forms, Ann. of Math. (2) 100 (1974), 249306. MR 0563090 (58 #27799)
 [9]
 James E. Humphreys, Linear algebraic groups, SpringerVerlag, New York, 1975. Graduate Texts in Mathematics, No. 21. MR 0396773 (53 #633)
 [10]
 N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 548. MR 0185016 (32 #2486)
 [11]
 Victor G. Kac, Infinitedimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219 (92k:17038)
 [12]
 Shrawan Kumar, KacMoody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, Birkhäuser Boston Inc., Boston, MA, 2002. MR 1923198 (2003k:22022)
 [13]
 Robert P. Langlands, Euler products, Yale University Press, New Haven, Conn., 1971. A James K. Whittemore Lecture in Mathematics given at Yale University, 1967; Yale Mathematical Monographs, 1. MR 0419366 (54 #7387)
 [14]
 Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, SpringerVerlag, Berlin, 1976. MR 0579181 (58 #28319)
 [15]
 Manish M. Patnaik, Geometry of loop Eisenstein series, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)Yale University. MR 2711828
 [16]
 Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR 1278263 (95b:11039)
 [17]
 Michael Rosen, Number theory in function fields, Graduate Texts in Mathematics, vol. 210, SpringerVerlag, New York, 2002. MR 1876657 (2003d:11171)
 [18]
 André Weil, Adeles and algebraic groups, Progress in Mathematics, vol. 23, Birkhäuser Boston, Mass., 1982. With appendices by M. Demazure and Takashi Ono. MR 670072 (83m:10032)
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Additional Information
KyuHwan Lee
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email:
khlee@math.uconn.edu
Philip Lombardo
Affiliation:
Department of Mathematics and Computer Science, St. Joseph’s College, Patchogue, New York 11772
Email:
plombardo@sjcny.edu
DOI:
http://dx.doi.org/10.1090/S000299472013060783
PII:
S 00029947(2013)060783
Received by editor(s):
February 8, 2011
Received by editor(s) in revised form:
August 15, 2012
Published electronically:
September 26, 2013
Article copyright:
© Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
